Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable hierarchy and exact N-soliton solutions exhibiting unusual accelerating motion. We show that both the deformed KdV and mKdV systems possess infinitely many gener...

We show that the complex modified KdV (cmKdV) equation and generalized nonlinear Schrödinger (GNLS) equation belong to the Ablowitz, Kaup, Newell and Segur or so-called AKNS hierarchy. Both equations do not follow from the action principle and are nonintegrable. By introducing some auxiliary fields we obtain the variational principle for them and study their canonical structures. We make use of...

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modi-ed Korteweg{de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectr...

We introduce a new integrable system hierarchy which is a restriction of the AKNS n×n hierarchy coming from an unusual splitting of the loop algebra. This splitting comes from an automorphism of the loop algebra instead of an automorphism of SL(n,C). It is known that the 2 × 2 KdV is the standard KdV hierarchy. We construct a gauge equivalence of this n× n KdV hierarchy with the Gelfand-Dickey ...

This is an introductory report concerning our recent research on Hamiltonian structures. We will discuss variational identities associated with continuous and discrete spectral problems, and their applications to Hamiltonian structures of soliton equations. Our illustrative examples are the AKNS hierarchy and the Volterra lattice hierarchy associated with semisimple Lie algebras, and two hierar...

Using the nonlinear constraint and Darboux transformation methods, the (m1, · · · ,mN ) localized solitons of the hyperbolic su(N) AKNS system are constructed. Here “hyperbolic su(N)” means that the first part of the Lax pair is Ψy = JΨx + U(x, y, t)Ψ where J is constant real diagonal and U = −U . When different solitons move in different velocities, each component Uij of the solution U has at ...

Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution are obtained for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the Korteweg-de Vries (KdV) hierarchy. It is shown that the linearization operators and the recursion operator which generates the hierarchy are commutable. Consequently, eigenfunctions of the linearization operators are precisely ...

Abstract We propose a systematic method to generalize the integrable Rosochatius deformations for finite dimensional integrable Hamiltonian systems to integrable Rosochatius deformations for infinite dimensional integrable equations. Infinite number of the integrable Rosochatius deformed higher-order constrained flows of some soliton hierarchies, which includes the generalized integrable Hénon-...